Subgroups — C₂×#11720290 #13139414

Order 1728.

594 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6226	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#11720290	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63561	—	3	1	no	maximal
6	C₂×#63382	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
9	C₂×#6226	—	2	1	no	—
10	C₂×#6226	—	2	1	no	—
11	C₂×#6226	—	2	1	no	—
12	C₂×#6226	—	2	1	no	—
13	C₂×#6226	C₄	1	2	yes	—
14	C₂×#6226	C₂²	1	2	yes	—
15	C₉×#208	—	2	1	no	—
16	C₉×#208	C₂²	1	2	yes	—
17	C₂×C₉×C₃⋊C₈	—	2	1	no	—
18	C₃×#3667	—	2	1	no	—
19	C₃×#3667	C₂²	1	2	yes	—
20	C₂×C₃×C₉⋊C₈	—	2	1	no	—
21	C₂²×#634	—	3	1	no	—
22	C₂×C₃×#208	—	3	1	no	—
23	C₂×C₃×#208	D₃	1	1	yes	characteristic
24	C₂²×Dic₁₈	—	3	1	no	—
25	C₂²×Dic₁₈	—	6	1	no	—
26	#63561	—	3	4	no	—
27	C₂×C₉×M₁₆	—	3	1	no	—
28	C₂×#3667	D₃	1	1	yes	characteristic
29	#63382	—	3	4	no	—
30	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₃⋊Dic₉	—	2	2	no	—
38	C₂×C₃⋊Dic₉	—	2	2	no	—
39	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
40	#6226	—	2	2	no	—
41	#6226	—	2	2	no	—
42	#6226	—	2	2	no	—
43	#6226	—	2	2	no	—
44	C₉×C₃⋊C₈	—	2	2	no	—
45	C₃×C₉⋊C₈	—	2	2	no	—
46	C₂×#824	—	9	1	no	—
47	C₂×#824	—	9	1	no	—
48	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃⋊Dic₃	—	3	2	no	—
50	C₂×#634	—	3	2	no	—
51	C₂×#634	—	3	2	no	—
52	C₂×#634	—	6	1	no	—
53	C₂×#634	—	6	1	no	—
54	C₂×#634	—	6	1	no	—
55	C₂×#634	—	6	1	no	—
56	C₃×#208	—	2	1	no	—
57	C₃×#208	—	3	2	no	—
58	C₃×#208	—	6	1	no	—
59	C₃×#208	C₂×D₃	1	2	yes	—
60	C₂×C₃×C₃⋊C₈	—	2	1	no	—
61	C₂×C₃×C₃⋊C₈	—	6	1	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂²×Dic₉	—	3	2	no	—
65	C₂²×Dic₉	—	6	2	no	—
66	C₂×Dic₁₈	—	3	2	no	—
67	C₂×Dic₁₈	—	3	2	no	—
68	C₂×Dic₁₈	—	6	2	no	—
69	C₂×Dic₁₈	—	6	2	no	—
70	C₂×Dic₁₈	—	6	2	no	—
71	C₂×Dic₁₈	—	6	2	no	—
72	C₂×Dic₁₈	—	6	2	no	—
73	C₂×Dic₁₈	—	6	2	no	—
74	C₂×Dic₁₈	—	6	1	no	—
75	C₂×Dic₁₈	—	6	1	no	—
76	C₂×Dic₁₈	—	6	1	no	—
77	C₂×Dic₁₈	—	6	1	no	—
78	C₉×M₁₆	—	3	2	no	—
79	C₉×M₁₆	—	6	1	no	—
80	C₂×C₈×C₉	—	6	1	no	—
81	#3667	—	2	1	no	—
82	#3667	C₂×D₃	1	2	yes	—
83	C₂×C₉⋊C₈	—	2	1	no	—
84	C₂²×C₃×C₉	—	2	2	no	—
85	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
92	C₃⋊Dic₉	—	2	4	no	—
93	C₂²×Dic₆	—	9	1	no	—
94	C₂²×Dic₆	—	9	1	no	—
95	C₂²×Dic₆	—	18	1	no	—
96	C₂×C₃×M₁₆	—	3	1	no	—
97	C₂×C₃×M₁₆	—	9	1	no	—
98	C₂×#208	—	3	1	no	—
99	C₂×#208	D₉	1	1	yes	characteristic
100	#824	—	9	4	no	—
101	#824	—	9	4	no	—
102	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
104	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
105	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
106	C₂×C₃²×C₄	#67	1	1	yes	characteristic
107	C₂×C₃²×C₄	#67	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃⋊Dic₃	—	3	4	no	—
110	C₂×C₃⋊Dic₃	—	6	2	no	—
111	C₂×C₃⋊Dic₃	—	6	2	no	—
112	#634	—	6	2	no	—
113	#634	—	6	2	no	—
114	#634	—	6	2	no	—
115	#634	—	6	2	no	—
116	C₂³×C₉	—	2	1	no	—
117	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
118	C₃×C₃⋊C₈	—	2	2	no	—
119	C₃×C₃⋊C₈	—	6	2	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
127	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
128	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
129	C₂×C₄×C₉	#67	1	1	yes	characteristic
130	C₂×C₄×C₉	#67	1	1	yes	characteristic
131	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
132	C₂×Dic₉	—	3	4	no	—
133	C₂×Dic₉	—	6	4	no	—
134	C₂×Dic₉	—	6	4	no	—
135	C₂×Dic₉	—	6	4	no	—
136	C₂×Dic₉	—	6	2	no	—
137	C₂×Dic₉	—	6	2	no	—
138	Dic₁₈	—	6	4	no	—
139	Dic₁₈	—	6	4	no	—
140	Dic₁₈	—	6	4	no	—
141	Dic₁₈	—	6	4	no	—
142	Dic₁₈	—	6	2	no	—
143	Dic₁₈	—	6	2	no	—
144	Dic₁₈	—	6	2	no	—
145	Dic₁₈	—	6	2	no	—
146	C₈×C₉	—	6	2	no	—
147	C₉⋊C₈	—	2	2	no	—
148	C₂×#101	—	27	1	no	Sylow
149	C₂×C₃×C₉	—	2	1	no	—
150	C₂×C₃×C₉	—	2	1	no	—
151	C₂×C₃×C₉	#101	1	2	yes	complemented
152	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
155	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
156	C₂²×Dic₃	—	9	2	no	—
157	C₂²×Dic₃	—	9	2	no	—
158	C₂²×Dic₃	—	18	2	no	—
159	C₂×Dic₆	—	9	2	no	—
160	C₂×Dic₆	—	9	2	no	—
161	C₂×Dic₆	—	9	2	no	—
162	C₂×Dic₆	—	9	2	no	—
163	C₂×Dic₆	—	18	2	no	—
164	C₂×Dic₆	—	18	2	no	—
165	C₂×Dic₆	—	18	2	no	—
166	C₂×Dic₆	—	18	2	no	—
167	C₂×Dic₆	—	18	2	no	—
168	C₂×Dic₆	—	18	2	no	—
169	C₂×Dic₆	—	18	1	no	—
170	C₂×Dic₆	—	18	1	no	—
171	C₂×Dic₆	—	18	1	no	—
172	C₂×Dic₆	—	18	1	no	—
173	C₂×Dic₆	—	18	1	no	—
174	C₂×Dic₆	—	18	1	no	—
175	C₂×Dic₆	—	18	1	no	—
176	C₂×Dic₆	—	18	1	no	—
177	C₃×M₁₆	—	3	2	no	—
178	C₃×M₁₆	—	6	1	no	—
179	C₃×M₁₆	—	9	2	no	—
180	C₃×M₁₆	—	18	1	no	—
181	C₂×C₃×C₈	—	6	1	no	—
182	C₂×C₃×C₈	—	18	1	no	—
183	#208	—	2	1	no	—
184	#208	—	3	2	no	—
185	#208	—	6	1	no	—
186	#208	C₂×D₉	1	2	yes	—
187	C₂×C₃⋊C₈	—	2	1	no	—
188	C₂×C₃⋊C₈	—	6	1	no	—
189	C₂²×C₃²	—	2	2	no	—
190	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
193	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
195	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
196	C₃²×C₄	C₂×#67	1	1	yes	characteristic
197	C₃⋊Dic₃	—	6	4	no	—
198	C₂²×C₉	—	2	4	no	—
199	C₂²×C₉	—	2	2	no	—
200	C₂²×C₉	—	2	1	no	—
201	C₂²×C₉	—	2	1	no	—
202	C₂²×C₉	—	2	1	no	—
203	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
206	C₄×C₉	—	2	1	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	2	1	no	—
209	C₄×C₉	—	2	1	no	—
210	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
213	C₄×C₉	C₂×#67	1	1	yes	characteristic
214	Dic₉	—	6	8	no	—
215	Dic₉	—	6	4	no	—
216	C₂²×Q₈	—	27	1	no	—
217	C₂×M₁₆	—	3	1	no	—
218	C₂×M₁₆	—	9	1	no	—
219	#101	—	27	4	no	—
220	C₃×C₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
221	C₂³×C₃	—	2	1	no	—
222	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
223	C₂³×C₃	#624	1	1	yes	characteristic
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
231	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
232	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
233	C₂×C₃×C₄	#611	1	1	yes	characteristic
234	C₂×C₃×C₄	#611	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
236	C₂×C₃×C₄	#624	1	1	yes	characteristic
237	C₂×C₃×C₄	#626	1	1	yes	characteristic
238	C₂×C₃×C₄	#626	1	1	yes	characteristic
239	C₂×C₃×C₄	#626	1	1	yes	characteristic
240	C₂×C₃×C₄	#626	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
242	C₂×Dic₃	—	9	4	no	—
243	C₂×Dic₃	—	9	4	no	—
244	C₂×Dic₃	—	18	4	no	—
245	C₂×Dic₃	—	18	4	no	—
246	C₂×Dic₃	—	18	4	no	—
247	C₂×Dic₃	—	18	2	no	—
248	C₂×Dic₃	—	18	2	no	—
249	C₂×Dic₃	—	18	2	no	—
250	C₂×Dic₃	—	18	2	no	—
251	Dic₆	—	18	4	no	—
252	Dic₆	—	18	4	no	—
253	Dic₆	—	18	4	no	—
254	Dic₆	—	18	4	no	—
255	Dic₆	—	18	2	no	—
256	Dic₆	—	18	2	no	—
257	Dic₆	—	18	2	no	—
258	Dic₆	—	18	2	no	—
259	Dic₆	—	18	2	no	—
260	Dic₆	—	18	2	no	—
261	Dic₆	—	18	2	no	—
262	Dic₆	—	18	2	no	—
263	C₃×C₈	—	6	2	no	—
264	C₃×C₈	—	18	2	no	—
265	C₃⋊C₈	—	2	2	no	—
266	C₃⋊C₈	—	6	2	no	—
267	C₂×C₃²	—	2	1	no	—
268	C₂×C₃²	—	2	1	no	—
269	C₂×C₃²	#824	1	2	yes	—
270	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
271	C₂×C₉	—	2	2	no	—
272	C₂×C₉	—	2	2	no	—
273	C₂×C₉	—	2	1	no	—
274	C₂×C₉	—	2	1	no	—
275	C₂×C₉	—	2	2	no	—
276	C₂×C₉	—	2	1	no	—
277	C₂×C₉	#824	1	2	yes	complemented
278	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
279	C₂×Q₈	—	27	2	no	—
280	C₂×Q₈	—	27	2	no	—
281	C₂×Q₈	—	54	1	no	—
282	C₂×Q₈	—	54	1	no	—
283	C₂×Q₈	—	54	1	no	—
284	C₂×Q₈	—	54	1	no	—
285	C₂²×C₄	—	27	2	no	—
286	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
287	M₁₆	—	3	2	no	—
288	M₁₆	—	6	1	no	—
289	M₁₆	—	9	2	no	—
290	M₁₆	—	18	1	no	—
291	C₂×C₈	—	6	1	no	—
292	C₂×C₈	—	18	1	no	—
293	C₂²×C₃	—	2	4	no	—
294	C₂²×C₃	—	2	2	no	—
295	C₂²×C₃	—	2	2	no	—
296	C₂²×C₃	—	2	1	no	—
297	C₂²×C₃	—	2	1	no	—
298	C₂²×C₃	—	2	1	no	—
299	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
300	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
301	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
302	C₂²×C₃	#3722	1	1	yes	characteristic
303	C₂²×C₃	#3722	1	1	yes	characteristic
304	C₂²×C₃	C₂×#624	1	1	yes	characteristic
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	1	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	2	1	no	—
309	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
310	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
311	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
312	C₃×C₄	C₂×#611	1	1	yes	characteristic
313	C₃×C₄	#3722	1	1	yes	characteristic
314	C₃×C₄	#3722	1	1	yes	characteristic
315	C₃×C₄	C₂×#626	1	1	yes	characteristic
316	C₃×C₄	C₂×#626	1	1	yes	characteristic
317	Dic₃	—	18	8	no	—
318	Dic₃	—	18	4	no	—
319	Dic₃	—	18	4	no	—
320	C₃²	C₂×#824	1	1	yes	characteristic
321	C₉	—	2	1	no	—
322	C₉	C₂×#824	1	1	yes	characteristic, complemented
323	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
324	Q₈	—	54	2	no	—
325	Q₈	—	54	2	no	—
326	Q₈	—	54	2	no	—
327	Q₈	—	54	2	no	—
328	C₂×C₄	—	27	4	no	—
329	C₂×C₄	—	54	2	no	—
330	C₂×C₄	—	54	2	no	—
331	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
332	C₂×C₄	#6192	1	1	yes	characteristic
333	C₂×C₄	#6192	1	1	yes	characteristic
334	C₂×C₄	#6195	1	1	yes	characteristic
335	C₂×C₄	#6195	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
337	C₈	—	6	2	no	—
338	C₈	—	18	2	no	—
339	C₂×C₃	—	2	2	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	—	2	1	no	—
342	C₂×C₃	—	2	1	no	—
343	C₂×C₃	—	2	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	2	no	—
347	C₂×C₃	#63382	1	2	yes	complemented
348	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
349	C₂×C₃	#63561	1	2	yes	—
350	C₂×C₃	C₂×#3722	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	#88445	1	1	yes	characteristic
353	C₂²	#88445	1	1	yes	characteristic
354	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
355	C₄	—	54	4	no	—
356	C₄	#88445	1	1	yes	characteristic
357	C₄	#88445	1	1	yes	characteristic
358	C₄	C₂×#6192	1	1	yes	characteristic
359	C₄	C₂×#6195	1	1	yes	characteristic
360	C₃	—	2	1	no	—
361	C₃	C₂×#63382	1	1	yes	characteristic, complemented
362	C₃	C₂×#63561	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	#11720290	1	2	yes	direct product factor
366	C₂	C₂×#88445	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88445
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#101
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #11720290	C₂	direct (N × H)
151. C₂×C₃×C₉	#101	semidirect
220. C₃×C₉	C₂×#101	semidirect
277. C₂×C₉	#824	semidirect
322. C₉	C₂×#824	semidirect
347. C₂×C₃	#63382	semidirect
361. C₃	C₂×#63382	semidirect
365. C₂	#11720290	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#6226	2
C₂×C₉×#208	2
C₂×C₃×#3667	2
#11720290	2
C₂×#63561	3
C₂×#63382	3

Subgroups — C2×#11720290 #13139414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720290 #13139414